T ransport and P ercolation in C omplex N etworks

1. Transportand Percolationin Complex Networks Guanliang Li Advisor: H. Eugene Stanley Collaborators: Shlomo Havlin, Lidia A. Braunstein, Sergey V. Buldyrev and José S. Andrade Jr.

2. Outline • Part I: Towards design principles for optimal transport networks G. Li,S. D. S. Reis, A. A. Moreira, S. Havlin, H. E. Stanleyand J. S. Andrade, Jr., PRL 104, 018701 (2010); PRE (submitted) Motivation: e.g. Improving the transport of New York subway system Questions: How to add the new lines?  How to design optimal transport network? • Part II: Percolation on spatially constrained networks D. Li, G. Li, K. Kosmidis, H. E. Stanley, A. Bunde and S. Havlin, EPL 93, 68004 (2011) Motivation: Understanding the structure and robustness of spatially constrained networks Questions: What are the percolation properties? Such as thresholds . . . What are the dimensions in percolation?  How are spatial constraints reflected in percolation properties of networks?

3. Part I: Towards design principles for optimal transport networks Grid network G. Li,S. D. S. Reis, A. A. Moreira, S. Havlin, H. E. Stanleyand J. S. Andrade, Jr., PRL 104, 018701 (2010); PRE (submitted) Shortest path length ℓAB =6 ℓ ~ L D. J. Watts and S. H. Strogatz, Collective dynamics of “small-world” networks, Nature, 393 (1998). Randomly add some long-range links: small-world network ℓ ~log L A B ℓAB =3 L How to add the long-range links?

4. Part I How to add the long-range links? Rich in short-range connections A few long-range connections Kleinberg model of social interactions J. Kleinberg, Navigation in a Small World, Nature 406, 845 (2000).

5. Part I Kleinberg model continued P(rij) ~ rij- where   0, and rij is lattice distance between node i and j • Steps to create the network: • Randomly select a node i • Generate rij from P(rij), e.g. rij= 2 • Randomly select node j from those 8 nodes on dashed box • Connect i and j Q: Which  gives minimal average shortest distance ℓ ?

6. Part I Optimal  in Kleinberg’s model Minimal ℓ occurs at =0 Without considering the cost of links d is the dimension of the lattice

7. Part I Considering the cost of links • Each link has a cost  lengthr (e.g. airlines, subway) • Have budget to add long-range links (i.e. total cost is usually  system size N) • Trade-off between the number Nland length of added long-range links From P(r) ~ r- : =0, r is large, Nl =  / r is small  large, r is small, Nl is large Q: Which  gives minimal ℓ with cost constraint?

8. Part I With cost constraint ℓ is the shortest path length from each node tothe central node Minimal ℓ occurs at =3

9. Part I With cost constraint ℓ vs. L (Same data on log-log plots) Conclusion: For ≠3, ℓ ~ L For =3, ℓ ~ (ln L) 

10. Part I With cost constraint Different lattice dimensions Optimal ℓ occurs at =d+1 (Total cost  ~ N) when N

11. Part I Conclusion, part I For regular lattices, d=1, 2 and 3, optimal transport occurs at =d+1 More work can be found in thesis (chapter 3): • Extended to fractals, optimal transport occurs at = df +1 • Analytical approach Empirical evidence 1. Brain network: L. K. Gallos, H. A. Makse and M. Sigman, PNAS, 109, 2825 (2011). df =2.1±0.1, link length distribution obeys P(r) ~ r - with =3.1±0.1 2. Airport network: G. Bianconi, P. Pin and M. Marsili, PNAS, 106, 11433 (2009). d=2, distance distribution obeys P(r) ~ r - with =3.0±0.2

12. Part II: Percolation on spatially constrained networks D. Li, G. Li, K. Kosmidis, H. E. Stanley, A. Bunde and S. Havlin, EPL 93, 68004 (2011) • P(r) ~ r - •  controls the strength of spatial constraint • =0, no spatial constraint  ER network •  large, strong spatial constraint  Regular lattice • Questions: • What are the percolation properties? Such as the critical thresholds, etc. • What are the fractal dimensions of the embedded network in percolation?

13. Part II The embedded network Start from an empty lattice Add long-range connections with P(r) ~ r - until k ~ 4 Two special cases: =0 Erdős-Rényi(ER) network  large Regular 2D lattice

14. Part II Percolation process Start randomly removing nodes Remove a fraction q of nodes, a giant component (red) exists Increase q, giant component breaks into small clusters when q exceeds a threshold qc, with pc=1-qc nodes remained small clusters giant component exists p pc

15. Part II Giant component in percolation =1.5 =2.5 =3.5 =4.5

16. Part II Result 1: Critical threshold pc 0.33

17. Part II Result 2: Size of giant component: M M ~ N in percolation : critical exponent =0.76

18. Part II Result 3: Dimensions M ~ N  in percolation • Percolation (p=pc): M ~ R df • Embedded network (p=1): N ~ R de • M ~ N df / de = df /de • Examples: • ER: df, de, but  =2/3 • 2D lattice: df=1.89, de=2,  =0.95 • So we compare  with df/de

19. Part II compare  with df/de In percolation Embedded network N

20. Part II Conclusion, part II: Three regimes K. Kosmidis, S. Havlin and A. Bunde, EPL 82, 48005 (2008)  ≤ d, ER (Mean Field) • > 2d, • Regular lattice Intermediate regime, depending on  Transport properties ℓ still show three regimes.

21. Summary • For cost constrained networks, optimal transport occurs at =d+1 (regular lattices) or df +1 (fractals) • The structure of spatial constrained networks shows three regimes: •  ≤ d, ER (Mean Field) • d <  ≤ 2d, Intermediate regimes, percolation properties depend on  •  > 2d, Regular lattice